Average Error: 58.5 → 0.0
Time: 22.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2812561 = 1.0;
        double r2812562 = 2.0;
        double r2812563 = r2812561 / r2812562;
        double r2812564 = x;
        double r2812565 = r2812561 + r2812564;
        double r2812566 = r2812561 - r2812564;
        double r2812567 = r2812565 / r2812566;
        double r2812568 = log(r2812567);
        double r2812569 = r2812563 * r2812568;
        return r2812569;
}
↓
double f(double x) {
        double r2812570 = x;
        double r2812571 = log1p(r2812570);
        double r2812572 = -r2812570;
        double r2812573 = log1p(r2812572);
        double r2812574 = r2812571 - r2812573;
        double r2812575 = 0.5;
        double r2812576 = r2812574 * r2812575;
        return r2812576;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

    \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]

  2. Simplified58.5

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(\frac{x + 1}{1 - x}\right)}\]
  3. Using strategy rm

  4. Applied log-div58.5

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(x + 1\right) - \log \left(1 - x\right)\right)}\]

  5. Simplified50.5

    \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\mathsf{log1p}\left(x\right)} - \log \left(1 - x\right)\right)\]
  6. Using strategy rm

  7. Applied log1p-expm1-u50.5

    \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(x\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(1 - x\right)\right)\right)}\right)\]

  8. Simplified0.0

    \[\leadsto \frac{1}{2} \cdot \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(\color{blue}{-x}\right)\right)\]

  9. Final simplification0.0

    \[\leadsto \left(\mathsf{log1p}\left(x\right) - \mathsf{log1p}\left(-x\right)\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))